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Lab 3: Differential Equations and Data
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:Author: Tim Reluga and Elissa Schwartz
:Date: Tuesday, June 14, 2011

Objectives
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* Use linear least squares to determine growth and
decay rates in linear models.

* Use nonlinear least squares to estimate parameters
for parameter estimation.

Procedures
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1) Open a browser to
http://www.math.psu.edu/treluga/yaleimmunology/lab3
This directory contains files and background used in this lab.

Oscillatory time-series data and nonlinear least squares
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We can start with something that's "simple".

The directory OSC contains a time series for a chemical species X
involved in a system of reactions with oscillatory dynamics.  Find
parameter values for the autocatalytic reactions

	0 -> X
	0 -> Y
	X -> 0  ( rate = 1 )
	2 X + Y -> 3 X  ( rate = 1 )

that fit this time-series.  You can download the COPASI file
so you don't have to enter things from scratch.  The data have 
been non-dimensionalized, so that two of the reaction rates 
are 1, so you need only fit the other two and the initial 
conditions.


Treatment of HIV patients
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2) Directory "HIV1" contains data files for two patients treated for
HIV around 1994.  These are from the paper Wei1995.pdf.  Use COPASI to
fit an exponential decay model to these data (linear least squares
would suffice, but fitting  the decay model is more natural in
COPASI).

* How do the rates you calculate compare to those calculated
in Wei'95?  How can you get your estimates to match theirs?

* What does your estimated decay rate tell us about the 
the immunological dynamics?


3) Directory "HIV2" contains data files for HIV treatment
patient with more frequent sampling.  Use COPASI and nonlinear
least squares to fit the model of Perelson1996.pdf to
these data.  Note that you have to fit both the parameters AND
the initial conditions for each patients.  (the
patient3.txt data is a little different from the first
two, and can be ignored)

	+----------------------------------------------+  
	| dT/dt = lambda - k T V - d T                 |
	| dI/dt = (1-epsilon_{RT}) k T V - delta I     |
	| dV/dt = (1-epsilon_{PI}) p I - c V           |
	| dW/dt =     epsilon_{PI} p I - c W           |
	+----------------------------------------------+  

* What does your estimated decay rate tell us about the 
the immunological dynamics?

* Why can this fit tell us more than the fit in (2)?

* Do these data and model convince you that the rate of viral
dynamics turnover is very fast (on the order a few hours) in HIV?